852HW10 - HOMEWORK ASSIGNMENT 10 PHYS852 Quantum Mechanics II Spring 2009 New topics covered partial waves scattering resonances 1 Hard-sphere

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Unformatted text preview: HOMEWORK ASSIGNMENT 10 PHYS852 Quantum Mechanics II, Spring 2009 New topics covered: partial waves, scattering resonances . 1. Hard-sphere S-wave scattering: Consider S-wave scattering from a hard sphere of radius a . Make the ansatz ψ ( r,θ,φ ) = e- ikr r- (1 + 2 ikf ( k )) e ikr r and show that it is an eigenstate of the full Hamiltonian for all r > a . Fit the value of f ( k ) to satisfy the boundary condition ψ ( a,θ,φ ) = 0. What is the partial amplitde f ( k )? What is the s-wave phase-shift δ ( k )? 2. Hard-sphere P-wave scattering: For P-wave scattering from a hard sphere of radius a , make the ansatz ψ ( r,θ ) = bracketleftbiggparenleftbigg 1 kr- i ( kr ) 2 parenrightbigg e- ikr + (1 + 2 ikf 1 ( k )) parenleftbigg 1 kr + i ( kr ) 2 parenrightbigg e ikr bracketrightbigg Y 1 ( θ ) , and show that it is an eigenstate of the full Hamiltonian for r > a . Again solve for the partial amplitude f 1 ( k ) by imposing the boundary condition ψ ( a,θ ) = 0. What is the phase-shift) = 0....
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This note was uploaded on 10/25/2010 for the course PHYSICS PHYS 851 taught by Professor Michaelmoore during the Fall '08 term at Michigan State University.

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